DLMF: SPECIAL FUNCTIONS IN THE 21ST CENTURYSCOPE OF COVERAGE • Methods (3 chapters) – Algebraic...

DLMF: SPECIAL FUNCTIONS IN THE 21ST CENTURY

Adri B. Olde DaalhuisMaxwell Institute and School of Mathematics,

University of Edinburgh, UK

THE

U NI V E R S I T

Y

OF

E D I N B U

RGH

THE DLMF• What led to its creation?

• Overview of its capabilities

• How was it created?

• The current state

• The future?!

Math Functions at NBS

• Math Tables Project (NY, 1938-46) – 37 volumes issued: trig, exp, log, etc

• NBS National Applied Math Lab (1947) – Institute of Numerical Analysis (UCLA) – Computation Lab, Statistical Engineering Lab (Washington)

• Applied Mathematics Series – AMS 1, Bessel Functions, 1948 – 1952 conference recommends compendium of tables – supported by NSF, NBS; began December 1956

3

(NBS = National Bureau of Standards,now NIST = National Institute of Standards and Technology)

11/08/2018, 20(57Ng and Shum Solicitors » Intellectual Property – Hong Kong – Copyright

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0

550

1100

1650

2200

1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2009

Increasing Trend of Citations to 1964 Handbook By Year, Every Third Year, 1971--2010

Preface The present volume is an outgrowth of a Conference on Mathematical

Tables held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Insti-tute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale com-puting machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist.

Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, 9f the increasing use of automatic computers. In the latter connection, the tables serve mainly for preliminary surveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable.

Consequently, the C"onference recognized that there was a pressing need for · a modernized version of the classical tables of functions pf Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare sucp a voiume and established an Ad Hoc Advisory Committee, .with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J . B. Rosser, H. C. Thacher, Jr., John Todd, C._B. Tompkins, and J. W. Tukey.

The primary aim has been to include a maximum of useful informa-tion within the limits of a moderately large volume, with particular atten-tion to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by the Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in compu-tation work, as well as by providing numerical methods which demonstrate the use and extension of the tables.

The Handbook was prepared under the direction of the late Milton Abramowitz, and Irene A. Stegun. Its success has depended greatly upon the cooperation of many mathematicians. Their efforts together with the cooperation of the Ad Hoc Committee are greatly appreciated. The par-ticular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized

It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions.

June 1964 Washington, D.C.

ALLEN V. AsTIN, Director

m

From the preface of AMS 55

• Original handbook: Mainly a list of tables. The formulas were added to make the tables more useful.


• New handbook: A book version with hardly any tables. The web version was included to introduce some extra features and information.



• The users of the old handbook are mainly physicists and engineers. The new handbook should also be useful for them.



• The users of the old handbook are mainly physicists and engineers. The new handbook should also be useful for them.

• Hence, we try to restrict ourselves to ‘useful’ formulas, that is, we do not immediately include all new results, and we do not include all possible formulas

THE DLMF PROJECT

• pre-1997: Requests for an update from Handbook users in the USA and beyond

• 1997: Project conception (ITL and PML) • 1999: Buy-in by NSF: $1.3 million

Associate Editors

• Richard A. Askey, University of Wisconsin

• Michael V. Berry, University of Bristol

• Walter Gautschi, Purdue University (resigned in 2002)

• Leonard C. Maximon, George Washington University

• Morris Newman, University of California

• Ingram Olkin, Stanford University

• Peter Paule, University of Linz

• William P. Reinhardt, University of Washington

• Nico M. Temme, Centrum Wiskunde Informatica

• Jet Wimp, Drexel University (resigned in 2001)

Chapter authors

• 2000: Authors selected, outline, first and second drafts, validators

• Frank Olver Authors

• 2010: Public Release: dlmf.nist.gov Publication: book with included cd

⟺

SCOPE OF COVERAGE

• Methods (3 chapters) – Algebraic and analytical methods – Asymptotic approximations – Numerical methods

• Mathematical Functions (33 chapters) – Elementary – Airy, Bessel, Legendre,… – Orthogonal polynomials – Elliptic integrals and functions – Combinatorics, number theory – Mathieu, Lamé, Heun, Painlevé, Coulomb,…

TYPICAL COVERAGE

Chapter 9

Airy and Related FunctionsF. W. J. Olver1

Notation 1949.1 Special Notation . . . . . . . . . . . . . 194

Airy Functions 1949.2 Di↵erential Equation . . . . . . . . . . . 1949.3 Graphics . . . . . . . . . . . . . . . . . . 1959.4 Maclaurin Series . . . . . . . . . . . . . . 1969.5 Integral Representations . . . . . . . . . 1969.6 Relations to Other Functions . . . . . . . 1969.7 Asymptotic Expansions . . . . . . . . . . 1989.8 Modulus and Phase . . . . . . . . . . . . 1999.9 Zeros . . . . . . . . . . . . . . . . . . . 2009.10 Integrals . . . . . . . . . . . . . . . . . . 2029.11 Products . . . . . . . . . . . . . . . . . . 203

Related Functions 2049.12 Scorer Functions . . . . . . . . . . . . . 2049.13 Generalized Airy Functions . . . . . . . . 2069.14 Incomplete Airy Functions . . . . . . . . 208

Applications 2089.15 Mathematical Applications . . . . . . . . 2089.16 Physical Applications . . . . . . . . . . . 209

Computation 2099.17 Methods of Computation . . . . . . . . . 2099.18 Tables . . . . . . . . . . . . . . . . . . . 2109.19 Approximations . . . . . . . . . . . . . . 2119.20 Software . . . . . . . . . . . . . . . . . . 212

References 212

1Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.Acknowledgments: This chapter is based in part on Abramowitz and Stegun (1964, Chapter 10) by H. A. Antosiewicz. The author

is pleased to acknowledge the assistance of Bruce R. Fabijonas for computing the numerical tables in §9.9, and of Leonard Maximon forwriting §9.16.

Copyright c� 2009 National Institute of Standards and Technology. All rights reserved.

193

WEBSITE VS. BOOK

• enhanced superset of the book with cutting-edge IT capabilities: – Color visualizations – Equation search (example: d^n/?^n) – Links

• Internal to symbol definitions, bib items, help… • External to online articles, reviews, software…

– Cut & paste tex, png, MathML – Sample applications

Kelvin’s shipwaveHankel function

Abdou Youssef, GWU and NIST

MathematicalKnowledge

Management

Bruce R. Miller

Introduction

arXMLiv

Math Semantics

Search

Conclusions

For example

Monodromy

TheDigital Library of

MathematicalFunctions

— Challenges —

Charles W. ClarkBruce R. Miller

Background &Motivation

AuthoringLATEX

LATExml

Examples

MathML

Presentation vs.ContentInterchange

Graphics

Tour

DLMF Technical Goals. . . decisions, decisions

! Book, fine-typesetting, many authors?⇒ LATEX.

! Searchable, richly linked, online text⇒ XML.

! Accessible, Reusable Mathematics?⇒ MathML!

! Interactive, ‘Honest’ Graphics?⇒ VRML! (X3D)

TheDigital Library of

MathematicalFunctions

— Challenges —

Charles W. ClarkBruce R. Miller

Background &Motivation

AuthoringLATEX

LATExml

Examples

MathML

Presentation vs.ContentInterchange

Graphics

Tour

So. . .

We decided to do it ourselves ⇒ LATExml.

Bruce R. Miller, NIST

More data is needed to makethis machine readable!

Why use the new handbook?


• An updated (corrected) version of the old handbook.



• We receive plenty of users feedback, and produce an update every 3 months.




• We give much more information: Sources and references to similar results.





• Mathml versions of the formulas, and probably other versions in the future.






• Hyperlinks!






• Hyperlinks!

• New results.

n! ∼ 2πn nne−n as n → ∞ .

Stirling approximation



Bleistein method Application: hypergeometric functions Stirling’s approximation

Stirling’s approximation

n! ⇠p2⇡n

⇣n

e

⌘n, as n ! 1.

Take n = 1 then 1! ⇠

p2⇡e

= 1.08...

Stirling’s grave in Greyfriars Kirkyard, Edinburgh. A 5 minute walk!Stirling’s grave in Greyfriars Kirkyard, Edinburgh


take n = 1



take n = 1

1! ≈ 0.922



take n = 1

1! ≈ 0.922


Exponential-asymptotics : the large variable is actually 2πn

1! ≈ 0.922

1! ≈ 0.922

Taking z = 1, K = 1, will give us RK(z) ≤ 0.10132...

User feedback

Olver notationWe give

Olver notationWe give

M(a, b, z) =Γ(1 + a − b)Γ(b)

2πiΓ(a) ∫(1+)

0eztta−1 (t − 1)b−a−1 dt

b ≠ 0, − 1, − 2,⋯ .instead of

Olver notation

Frank Olver : “Thank god that the Bessel functions are defined correctly.”

We give

M(a, b, z) =Γ(1 + a − b)Γ(b)

2πiΓ(a) ∫(1+)


b ≠ 0, − 1, − 2,⋯ .instead of

Olver notation


We give

M(a, b, z) =Γ(1 + a − b)Γ(b)

2πiΓ(a) ∫(1+)


b ≠ 0, − 1, − 2,⋯ .instead of

We give

Olver notation


We give

M(a, b, z) =Γ(1 + a − b)Γ(b)

2πiΓ(a) ∫(1+)


b ≠ 0, − 1, − 2,⋯ .instead of

We give

2F1 (a, bc

; z) =Γ(c)Γ(c − a − b)Γ(c − a)Γ(c − b) 2F1 ( a, b

a + b − c + 1; 1 − z) + (1 − z)c−a−b Γ(c)Γ(a + b − c)

Γ(a)Γ(b) 2F1 ( c − a, c − bc − a − b + 1

; 1 − z)but might include the more natural alternative version in future updates:

Unique visitors Visits Page downloadsSince 2010 1.6M 13.2M

In 2014 342K 2.7MMay 2015 21K 35K 349K

Most popular chapters: Bessel, gamma, confluent hypergeometric

07/23/2018

3

3. Findings In calendar year 2017, citations to the old Handbook (2,662 citations) outpaced citations to the new Handbook (653 citations) by a large margin. However, as in the previous studies, old Handbook citations continue a slow but steady decline. The overall percentage of citations to the old Handbook declined by 3% compared to 2016 (from 83% to 80% of total Handbook citations). Citations to the new Handbook, while still comparatively small, increased in 2017 by 16% over the 2016 citation count (from 565 to 653 citations). Figures 1 and 2 show the total number of citations per year for the old and new Handbooks, and the percentage of total citations per year for each Handbook.

Figure 1. Number of citations to the old and new editions of the Handbook of Mathematical Functions, 2008–2017

2,647 2,660 2,618 2,563 2,511 2,460 2,421 2,354 2,700 2,662

25 101 214 311 374 450

565 653

-

500

1,000

1,500

2,000

2,500

3,000

3,500

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Num

ber o

f Cita

tions

Calendar Year

New Handbook

Old Handbook

• Adri B. Olde Daalhuis, Mathematics Editor• Daniel W. Lozier, General Editor• Barry I. Schneider, General Editor• Howard Cohl, General Editor• Ronald F. Boisvert, Editor at Large• Charles W. Clark, Physical Sciences Editor• Bruce R. Miller, Information Technology Editor• Bonita V. Saunders, Visualization Editor

• Frank W. J. Olver served as Editor-in-Chief and Mathematics Editor for the DLMF project from its beginning until his death on April 23, 2013.

Editorial Board

The current state

The Current State

• The project is still very alive!

The Current State


• We receive several dozens of queries/error reports per year. Have a look at the errata page.

The Current State



• Once a year we investigate recent publications to try to find results that could be relevant for the DLMF.

The Current State




• We have now a list of Associate Editors who will be responsible for the chapters.

The Current State





The Current State

• Mark Ablowitz• George Andrews• Michael Berry• Annie Cuyt• Mourad Ismail

• James Pitman• Bill Reinhardt• Simon Ruijsenaars• Nico Temme• Stephen Watt

New list of Senior Associate Editors





The Current State




• We are enlarging the chapters: OP and Painlevé transcendents





The Current State





• A chapter on Orthogonal Polynomials of Several Variables is being created.





The Current State






• More new chapters (Computer Algebra?)





The Current State






• More new chapters (Computer Algebra?)

The numbering is still an issue,because we will have to add formulas in-between otherformulas

DLMF SO FAR

DLMF SO FAR• Selection of formulas chosen and checked by experts


• Original sources or short derivation are included



• Internal links to symbol definitions, bib items, help…




• External links to online articles, reviews, software…





• LaTeX and MathMl versions are included (should Maple/Mathematica versions be included?)






• Unique web links for each formula







• Powerful search capabilities








• We try to stay up-to-date, and are now creating new chapters








• We try to stay up-to-date, and are now creating new chapters

• Any feedback?

DLMF: SPECIAL FUNCTIONS IN THE 21ST CENTURYSCOPE OF COVERAGE • Methods (3 chapters) – Algebraic...

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